In the TBA literature, one typically chooses a parametrisation that guarantees symmetry of the differential scattering phase. As discussed in [1]}, the dressing operation in general is different for quantities \(h_n(\eta )\)  that are scalar fields, and those that are vector fields such as \(P^{\prime }(\eta )\) : the differential scattering phase appears with opposite order of its arguments. This guarantees that the dressing operation preserves the transformation property. But with the symmetric choice, the dressing has the same form in both case.
